Binomial Theorem

The Binomial Theorem is a fundamental theorem that provides a formula for expanding the powers of a binomial expression.

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Let x and y be two variables and n be non-negative interger then the binomial theorem for positive index n states that,

(a+x)n = C(n,0)an + C(n,1)an-1x + C(n,2)an-2x2 +…..+ C(n,n)xn

⁘ General Term of Binomial expansion of (a+x)ⁿ

The general term of the binomial expansion (a+x)ⁿ is denoted by t_r+1 and given by:

tr+1 = C(n,r) an-rxr

Note: if the expansion is (a-x)ⁿ , then its general term is given by:

tr+1 = C(n,r) an-rxr .(-1)r = (-1)r C(n,r) an-rxr

⁘ Middle Term

• When we expand (a+x)ⁿ , then we always get n+1 term in the expansion of (a+x)ⁿ.
• Therefore, if n is even then (n+1) will be odd and the expansion contains single middle term.
• Similarly, if n is odd then n+1 will be even and the expansion contains two middle terms.